"Is it correct to assume that 'gyroscopic precession' will always have a phase offset of 90-degrees, if the true precession that you have referred to is not considered?"
Yes, for small perturbations.
If the applied force is too big, relative to the mass and rpm, then the result is much more dynamic than merely a 90 degree apparent lag. If you set up the experiment of a bicylcle wheel spinning horizontally, hanging on a string, a small rap on the wheel will demonstrate the 90 degree phase angle. Hit it too hard and you start noticing some gyrating movements of the spin axis.
Those gyrations are predictable and are a result of conservation of angular momentum, but to describe them without pictures and without vector calculus would be challenging.
I needed only to check online dictionaries to find a number of definitions of precession that mentioned going before or forward. In that light, I guess the use of the word precession is not wrong.
Matthew.